Analogue device especially for the computation and adjustment of trilateration nets



Feb. 23, 1965 H. s. JERI 3,170,626

mums-us nsvzcs ESPECIA FOR THE PUTATION AND ADJUSTMENT 0F LATERATION NETS Filed Feb. 26, 1960 2 Sheets-Sheet l INVENTOR. HAN5 GEORG JE RIE A TTORNEYS 3,1 70,62 6 JUSTMENT H. G. JERIE IALLY FOR THE COMPUTATION AND AD 0F TRILATERATION NETS 2 Sheets-Sheet 2 FIG.3

INVHVTOR. HANS GEORG JERlE ATTORNEYS United States Patent 3,170,626 ANALOGUE DEVICE ESPECIALLY FOR THE COMPUTATION AND ADJUSTMENT 0F TRI- LATERATION NETS Hans Georg Jerie, Freyung 6/11, Vienna 1, Austria Filed Feb. 26, 1960, Ser. No. 11,198 7 Claims. (Cl. 235-61) The invention relates to mechanical means to be used as an analogue device and to a method for solving equations which proceed from the method of least squares, more in particular for the adjustment of trilateration networks.

The adjustment of trilateration networks includes the following problem:

Between a number of points in a surface, some of which have an exactly known position and some of which have an unknown position, distances are measured by means of some known method (for example by an electronic distance measuring method), more distances being measured than is strictly necessary for the calculation of the positions of the said points. The assembly of said distances is called a distance network or trilateration network. Since all the distances measured contain unavoidable errors, the most probable position has to be deter- -mined by an adjustment calculation with the aid of the method of the least squares.

In mathematically solving the problem by means of the method of the least squares which has been frequently used up till now, very complicated systems of linear equations have to be solved, in particular when the eo-ordinates of a great many points have to be determined, which often gives rise to great difiiculties.

The object of the invention is to provide a method and a mechanical means with which the said adjustment can be accomplished in a much shorter time which need not be operated by highly qualified personnel.

Another object of the invention is to provide a mechanical analogue device in which the said adjustment is obtained by making use of the known analogy between a mechanical elastic system and the adjustment theory.

Other objects of the invention will appear from the description below. My analogue device comprises:

(1) A base member, such as a table, having a smooth surface which is preferably covered with drawing paper or graph paper and which can be subjected to a vibratory or shaking movement. Advantageously a vibration device driven by an electric motor is mounted onto said base member. However, in some cases the base member may be shaken by hand.

On said base member, preferably covered with graph paper, the co-ordinates of the known points are plotted I in a certain scale (scale m,, the mapping scale).

(2) A number of marking devices which represent the various points, such as studs (rods provided with an axial bore and, at the lower side with a supporting flange), said studs being known per se in the field of art. Pins or needles can be put in the said bores in order either to pin the studs onto' the positions plotted or to mark the positions of the points obtained' (3) A number of elements which are hereinafter called distance rods, each of which comprises va rigid rod provided with two fixing elements which are slidable thereon, one of which (the free one) is fixable in an arbitrary position by means of agset screw, the other one being bound elastically on both sides byimeans of a pair of easily replaceable elastic means such .as helical springs. Each of said fixing elements is adapted to be mounted on one of said marking devices.

(4) One or more auxiliary devices which are hereinafter called setting devices, and which are employed to set the free fixing elementof a distance rod at a prede- 3,170,626 Patented Feb. 23, 1965 ice termined distance from the equilibrium position of the elastically bound fixing element in order that they may represent the end-points of a certain distance, in particular when each one is mounted onto a stud of the assembled network.

To illustrate the invention, an embodiment of the invention is shown in the drawing, in which:

FIG. 1 is an upper view of four studs with distance rods mounted therebetween;

FIG. 2 is a side view of one of the studs of FIG. 1;

FIG. 3 is a vertical cross-section of a stud, along the line IIIIII in FIG. 4;

FIG. 4 is a bottom view of a marking device;

FIG. 5 is an upper view of a setting device, and

FIG. 6 is a side view of the device of FIG. 5.

The marking devices each contain a stud 1 provided with an axial bore 2 and with a flange-like supporting member 3 which has an elevated central part 4 and which is provided on its lower outer side with a rim 5.

The elevated part is not massive, but it contains supporting ribs 6 as shown in FIGS. 3 and 4.

The marking devices, which represent the points of the network, are linked together by distance rods which are constructed of rigid rods 7 and, slidingly arranged thereon, an adjustable fixing element 8 and an elastically bound fixing element 9. Both of said elements are preferably constructed of a synthetic resin material, such as a polyamide or a polyester and are provided with gaps 10 which are adapted to slide along the studs 1 and to be arranged thereon and removed therefrom in a side-way direction by means of slightly elastic flaps 11. The adjustable fixing element 8 contains a set screw 12. The elastically bound fixing element is arranged between two helical springs 13 which are closed in between two removable setting elements 14 provided with set screws 15. In general the setting elements are adjusted so as just to ease the springs.

FIG. 1 shows a part of a distance network. In general, the more distance rods are joining in one point (on one stud) the more precise the position of said point is determined.

FIG. 2 also shows a part of the base member 16 provided with graph paper 17.

In the setting device of FIGS. 5 and 6, a bar 20 is provided with two heads 21 and 22.

A strip of metal 23, provided with a scale calibration 24, is mounted between said heads. A setting element 25 is slidingly mounted on the bar 20.

The head 21 and the element 25 both contain a cylindrical-pin 26 having the same diameter as the studs 1 so that both fixing elements of a distance rod can be placed on said pins in order toset the adjustable fixing element 8 at a predetermined distance from the equilibrium position of the elastically bound fixing element 9 which position is fixed by tightening the set screw 12. The setting element 25 is provided with a set screw 27 and with a pointer 28 which serves to read the distance between both pins on they scale calibration24.

My method comprises the following steps:

(a) The zero position of an elastic system, which is an analogue of the adjustment problem to be solved, is determined.

To this end, the co-ordinates of the known points are plotted on the base member, preferably covered with graph paper, whereupon the marking elements (studs) are pinned onto said plotted positions.

Then, a distance rod is prepared for each distance measured, in such a way that the elastic constant of the elastical means used, such as a pair of springs, is proportional to the weight of the distance measurement concerned and that the. distance between both fixing elements of. the rod equals the distance measured in the mapping scale (m The latter operation is carried out with the aid of the setting devices mentioned above.

As is known in the art, the weight of a measurement is inversely proportional to the square of the mean square error of said measurement.

Now, all of the distance rods with the fixed studs of the known points and with the freely islidable studs of the unknown points are arranged on the base member to form an assembly which is an analogue of the trilateration network. After shaking or vibrating the table, the zero positions of the unknown points, thus obtained, are marked on the base member, preferably covered with graph paper by pinning needles through the said studs.

(b) The diiferences (discrepancies) between the values of the distances measured (in the field) and of the distances computed from the approximate values of the coordinates of the points (which approximate values are obtained by any arbitrary method) are calculated. Such a method is for instance the scaling of the zero-position (see above) of the unknown points.

A method which gives more exact results is their cal culation from the distances measured.

(c) The said discrepancies are introduced into the rods, in such a way, that the distances between the fixing elements are modified by values corresponding to said discrepancies in an appropriate scale, the correction scale (m which is larger than the mapping scale (m The distance between the fixing elements d can now be expressed by the following formula:

in which Dir; is the real distance as it is measured in the field and AD the discrepancy computed for said distance.

When the distance rods thus modified are again ar- 4 of shortness is restricted to the data of two definite points (point 1 and point 2) of a network. The distance measured 1-2 amounts to 3967.24 meter. The plotting scale is l:30,000 (so that the distance 1-2 to be plotted is 132 millimeter) Tables A and B show the results of the measurements and calculations of the co-ordinates and distances respectively of points 1 and 2. The Roman ciphers (I, II, III, IV) denote the iteration steps. Table A shows, for each iteration step, the approximate co-ordinates, the corrections of the co-ordinates (in mm.), obtained from the analogue device and thereafter the corresponding correctness of the real co-ordinates in meters. The addition of the co-ordinate correctness to the approximate coco-ordinates yields the corrected approximate co-ordinates.

In Table B, the approximate co-ordinates of the endpoints of the distance concerned are given for each iteration step, as well as the co-ordinate diiference calculated by the abstraction of said co-ordinates and beside it is listed the distance D calculated from said difference. This is compared with the measured value of the distance D (in meters) and thus the discrepancy AD is obtained. After choosing the appropriate correction scale, the discrepancy Ad is expressed in mm. and, finally, the corrected distance d which is to be introduced into the assembly. Corresponding data are shown for the further iteration steps. The mapping scale is chosen so that the shortest distance a will not be less than about 120 mm.; the cor rection scale is chosen so, that the highest of the discrepancies Ad is about 20 to 25 mm. The correction scales in this example are:

I. 1:1,000 II. 1:100 III. 1:10

(A) CO-ORDINATES AND CO-ORDINATE CORRECTIONS OF THE POINTS Point 1 P0int2 X Y X Y Mm Meter Mm. Meter Mm. Meter Mm. Meter II 443 040. 00 104 287.00 442 51s. 00 100 486. 00 +13 +1.30 -14 -1. 2s -2. 30 -22 -2. 20

ranged into an assembly and the. table is vibrated so that DISTANCES AND THEIR DISCREPANCIES the assembly gets in an equilibrium position as to the 1 4:. elastic forces of the rods, the iieely shdable studs are DISTANCE H Meter shifted over a certain d1stance with regard to their D/d (marked) zero position. The new equilibrium position X Y 3967 24 132 of the studs 'is marked on the base member, preferably covered with grap p p y Pinning medles thl'gugh I 1 443 680.00 104 200. 00 D 3048.10 the axial bores of the studs. 2 540-00 190 480-00 1 +1905 The magnitudes of the displacements of the slidablc 14MB 3 780300 d 151 studs are measured on the base (in the correction scale II g3 2st). 104 207. 00 P 3005. m They correspond to the required corrections of the 13%00 $188 zii 54 approximate co-ordinates of the unknown points, which 5 In r 1 5 6 191 28 67 D3 correctionsare the same as those which could have been 2 44251570 b 31 3 a 5 3: calculated in a very cumbersome way by means of the 1 3 -60 3 801- d 107 method of the least squares. If necessary, the operation v 1 443 650,13 285.64 4 1 396131 described (steps b and c) is repeated several times, using 442 515-32 190 483-96 ADt/ad -00? 1 134.30 3 801.68 0 a larger correction scale each time, until with the 1terat1on 70 method described the aim. of the adjustment treatment has been reached.

Example The proof that the corrections to the approximate coordinates are identical to those formed by the method of the least squares, can be given by using a well-known mechanical law. According to this law the equilibrium of an elastic system is reached when the sum of all the static energies is at a minimum. The energy, needed for displacing an elastically bound fixing element with respect to the other one of the same distance rod is proportional to the square of this displacement multiplied by the elasticity coefficient of the springs used. As such a displacement corresponds to the correction of the distance observation, and asas assumedthe elasticity coeificient is proportional to the weight of the observation as used for the corresponding numerical treatment the sum of the energies can be described with the expression g, v, v, =minimum where g, denotes the weight of observation at as well as the elasticity coefiicient of the springs, v, denotes the correction of the observation at, as well as the displacement of the fixing element. Thereby it is proved that the equilibrium of the elastic system corresponds to the condition for the adjustment according to the method of the least squares.

What I claim is:

1. A mechanical analogue device for adjusting trilateration nets comprising a base member having a smooth fiat surface, marking devices slidably arranged on said smooth surface, marking devices atfixed to said smooth surface denoting known points, said marking devices provided with studs standing perpendicular to said smooth surface, distance rods, each of said distance rods having a first fixing element slidably mounted on said rod, a fixing means to fasten said fixing element to said rod in a predetermined position, two setting elements slidably mounted on said rod, fixing means to fasten said setting elements to said rod in predetermined positions, assecond fixing element slidably mounted on said rod between said setting elements, two elastic means deformable in longitudinal direction of said-rod, each of said elastic means abutting said second, fixing element and'one of the setting elements, said fixing elements further comprising adjacentto said rod means adapted to connect said fixing elementsfreely rotatably to said studs of said marking devices in sucha be arranged into an assembly in which the known points are represented by said marking devices afiixed to said a manner that said marking devices and distance rods can 7 smooth surface and the unknown pointsiare represented by said marking devices slidably arranged on said-smooth fixing element to said rod in a predetermined position,

two setting elements slidably mounted on said rod, fixing 'surface, which assembly Jwill compensate for the errors in measurement of the distances between the various points 5. The method for compensating for errors in measurement of distances in establishing a trilateration network which comprises the steps of plotting the co-ordinates of known points in said network in a mapping scale, elastically connecting the known points withthe unknown points in proportion to the distances previously measured whereby the distance between the known and unknown points can vary depending upon an elastic stress set up due to errors in measurement of the distances and whereby said elastic stress is proportional'to the errors in measurement of the distances measured, bringing the trilateration network assembly into a state of static equilibrium as to the elastic stresses, plotting the first co-ordinates of the unknown points, thus establishing the zero position of the trilateration network, calculating the discrepancies between the real distances measured and the V distances computed from the first co-ordinates of said unknown points, elastically reconnecting the known points with the unknown points in proportion to the distances previously determined as modified by amounts which correspond to said. discrepancies proportional to a first correction scale which is larger than said mapping scale, w iereby thedistance between the known and unknown points can vary depending upon an elastic stress set up due to errors in measurement of the distance measured as modified by said discrepancy correction, re-bringing the trilateration network assembly into a state of static equilibrium as to the elastic stresses and plotting the second co-ordinates of the unknown points, thus establishing modifications of the co-ordinates of the unknown points, which modifications correspond to the corrections of the measured distances.

6. The method of claim 5 wherein a further correction is made by the steps of calculating thediscrepancies between the real distances measured and the distances computed from the secondco-ordinates of said unknown points, elastically reconnectingthe known points with the unknown ,pointsin proportion to the distances previously determined-as modified by amounts which correspondto said discrepancies proportional to a second correction scale which is larger than said first correction scale, whereby the distance between the known and unknown.

points can vary depending ,upon an elastic stress set up due to'errors in measurement of the distance measured as modified by said discrepancy correction, re bringing the trilateration network assembly into a state of static equilibrium as to the elastic stresses and plotting the third co-ordinates of the unknownpoints, thus establishing modifications of the co-ordinates of the unknown points, which modifications correspond tothe corrections ofithe measured distances, and repeating the above steps utilizing a larger correction scale each time and each time means to connect said setting elements to said rod in f predetermined positions, a second fixingelement slidably mounted on said rod between said setting elements, two elastic means each of whichabuts one of the said setting elements at one end and the second fixing element at its other end, said fixing elements further comprising means adapted to connect said elements freely rotatably to ver tical posts of studs. 5 v i 3. A mechanical unit according to. claim 2, characterized in that said elastic means consistsof a helicoidal introducing the corrections established the previous time,

until the required precision of the co-ordinates of the unknown points is established.

7. The method of claim 6 wherein each correction scale is about 10 times larger than the previous correcr tion scale, a

References Cited in the file ofthispatent UNITED STATES PATENTS 522,053' Gates June 26, 1894 1,965,098 Eaton July 3, 1934 2,178,293 Wogeck Oct. 31,1939 2,182,707 Shipman Dec. 5, 1939 2,221,872 King et al. 1940 2,493,786 Swift 1950- 2,537,718 Trorey .Ian. 9, 19 51 2,693,033 Acker et al. Nov. 2, 1954 a FOREIGN PATENTS Australia July 6, 

5. THE METHOD FOR COMPENSATING FOR ERRORS IN MEASUREMENT OF DISTANCES IN ESTABLISHING A TRILATERATION NETWORK WHICH COMPRISES THE STEPS OF PLOTTING THE CO-ORDINATES OF KNOWN POINTS IN SAID NETWORK IN A MAPPING SCALE, ELASTICALLY CONNECTING THE KNOWN POINTS WITH THE UNKNOWN POINTS IN PROPORTION OF THE DISTANCES PREVIOUSLY MEASURED WHEREBY THE DISTANCE BETWEEN THE KNOWN AND UNKNOWN POINTS CAN VARY DEPENDING UPON AN ELASTIC STRESS SET UP DUE TO ERRORS IN MEASUREMENT OF THE DISTANCES AND WHEREBY SAID ELASTIC STRESS IS PROPORTIONAL TO THE ERRORS IN MEASUREMENT OF THE DISTANCE MEASURED, BRINGING THE TRILATERATION NETWORK ASSEMBLY INTO A STATE OF STATIC EQUILIBRIUM AS TO THE ELASTIC STRESSES, PLOTTING THE FIRST CO-ORDINATES OF THE UNKNOWN POINTS, THUS ESTABLISHING THE ZERO POSITION OF THE TRILATERATION NETWORK, CALCULATING THE DISCREPANCIES BETWEEN THE REAL DISTANCES MEASURED AND THE DISTANCES COMPUTED FROM THE FIRST CO-ORDINATES OF SAID UNKNOWN POINTS, ELASTICALLY RECONNECTING THE KNOWN POINTS WITH THE UNKNOWN POINTS IN PROPORTION TO THE DISTANCES PREVIOUSLY DETERMINED AS MODIFIED BY AMOUNTS WHICH CORRESPOND TO SAID DISCREPANCIES PROPORTIONAL TO A FIRST CORRECTION SCALE WHICH IS LARGER THAN SAID MAPPING SCALE, WHEREBY THE DISTANCE BETWEEN THE KNOWN AND UNKNOWN POINTS CAN VARY DEPENDING UPON AN ELASTIC STRESS SET UP DUE TO ERRORS IN MEASUREMENT OF THE DISTANCE MEASURED AS MODIFIED BY SAID DISCREPANCY CORRECTION, RE-BRINGING THE TRILATERATION NETWORK ASSEMBLY INTO A STATE OF STATIC EQUILIBRIUM AS TO THE ELASTIC STRESSES AND PLOTTING THE SECOND CO-ORDINATES OF THE UNKNOWN POINTS, THUS ESTABLISHING MODIFICATIONS OF THE CO-ORDINATES OF THE UNKNOWN POINTS, MEASURED DISTANCES. 